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The conjugate Dirichlet kernel, and the periodic principal-value formula

Statement

Assume Countable Choice, and work on $T = R / Z$ with the conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus. Let $f \in L^{2} (T; C)$ and let $N \geq 1$ . Put

$K_{N} (t) : = 2 \sum_{k = 1}^{N} \sin (2 π k t) = \frac{\cos (π t) - \cos ((2 N + 1) π t)}{\sin (π t)} (t \notin Z),$

the conjugate Dirichlet kernel, and write $C S_{N} f = \sum_{0 < ∣ k ∣ \leq N} (- i sgn k) \hat{f} (k) e_{k}$ for the conjugate partial sum. Then:

$C S_{N} f$ is the convolution of $f$ with $K_{N}$ : for almost every $x$ , $C S_{N} f (x) = \int_{0}^{1} K_{N} (t) f (x - t) d t$ .
Extend $C$ to $L^{2} (T; C)$ by the square-summable coefficient family $(- i sgn k \hat{f} (k))_{k \in Z}$ ; the resulting class $C f$ is the $L^{2}$ limit of the partial sums $C S_{N} f$ .
If a representative of $f$ is $C^{1}$ on an open interval containing $x$ , then

$C f (x) = \lim_{ε ↓ 0} \int_{ε < ∣ t ∣ < 1 / 2} f (x - t) \cot (π t) d t,$

the limit existing for almost every such $x$ , and being the symmetric principal value about the singularity.

The finite kernel $K_{N}$ is not itself a cotangent truncation: $K_{N}$ equals $\cot (π t)$ minus the oscillatory remainder $\cos ((2 N + 1) π t) / \sin (π t)$ , and only the limit $N \to \infty$ of the convolutions recovers the principal value.

Facts & Assumptions

Given: Countable Choice, $f \in L^{2} (T; C)$ , $N \geq 1$ , and the characters $e_{k} (x) = e^{2 π i k x}$ .

[F1]

$C$ is defined on trigonometric polynomials coefficientwise by $\hat{C g} (k) = - i sgn (k) \hat{g} (k)$ , it is complex-linear, kills constants, and preserves real-valuedness. Conjugate function on the circle

[F2]

Fourier coefficients, partial sums $S_{N} f$ , characters, and the torus convolution $(f * g) (x) = \int_{0}^{1} f (x - t) g (t) d t$ are as defined there, and the torus integral is invariant under the reflections used below. Period-one Fourier coefficients, partial sums, and convolution on the torus

[F3]

The Dirichlet kernel is $D_{N} (t) = \sum_{∣ k ∣ \leq N} e_{k} (t)$ . Dirichlet and Fejer kernels

[F4]

For one-period integrable $f$ , $S_{N} f (x) = \int_{0}^{1} f (x - t) D_{N} (t) d t = (f * D_{N}) (x)$ for every $x$ . Fourier partial sums are Dirichlet convolutions

[F5]

For $N \geq 1$ and $x \notin 2 π Z$ , $\sum_{n = 1}^{N} \sin (n x) = \frac{\cos (x / 2) - \cos ((N + 1 / 2) x)}{2 \sin (x / 2)}$ . Finite sums of the sine harmonics

[F6]

Parseval: $∥ f ∥_{2}^{2} = \sum_{k \in Z} ∣ \hat{f} (k) ∣^{2}$ in the finite-subset-supremum sense, so the tails over ${∣ k ∣ > N}$ tend to $0$ . The Parseval identity for Fourier series

[F7]

Every square-summable coefficient family in $ℓ^{2} (Z; C)$ is the Fourier coefficient family of a unique $L^{2}$ class, realized as the $L^{2}$ limit of its symmetric partial sums. Riesz–Fischer: the Fourier coefficient map is onto the space of square-summable families

[F8]

Riemann-Lebesgue: if $g$ is integrable on one period then $\hat{g} (k) \to 0$ as $∣ k ∣ \to \infty$ . Riemann-Lebesgue lemma for Fourier coefficients

[F9]

Norm-convergent sequences in $L^{2}$ have subsequences converging almost everywhere to a representative of the limit. Complex Lp completeness and almost-everywhere subsequences

[F10]

The real mean value theorem bounds the increment of a real $C^{1}$ function by the supremum of its derivative times the interval length. Applied separately to the real and imaginary parts, it gives $∣ g (t) - g (0) ∣ \leq C ∣ t ∣$ for a complex $C^{1}$ function on a compact interval about $0$ , with $C = ∥ Re g^{'} ∥_{\infty} + ∥ Im g^{'} ∥_{\infty}$ . The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$

[F11]

Dominated convergence. Dominated convergence

[F12]

Cosine addition formula: $\cos (A + B) = \cos A \cos B - \sin A \sin B$ . The addition formulas for sine and cosine

Proof

technique · direct

1.1F1F3F5

By [F1] and [F3], $K_{N} : = C D_{N}$ is the trigonometric polynomial with coefficients $- i sgn (k)$ on $0 < ∣ k ∣ \leq N$ and $0$ elsewhere, so $K_{N} (t) = \sum_{0 < ∣ k ∣ \leq N} (- i sgn k) e_{k} (t) = 2 \sum_{k = 1}^{N} \sin (2 π k t)$ , using $e_{k} - e_{- k} = 2 i \sin (2 π k t)$ . Applying [F5] with $x = 2 π t$ gives $2 \sum_{k = 1}^{N} \sin (2 π k t) = \frac{\cos (π t) - \cos ((2 N + 1) π t)}{\sin (π t)}$ for $t \notin Z$ , while $K_{N} (0) = 0$ ; in particular $K_{N}$ is odd, one-periodic, and $\int_{- 1 / 2}^{1 / 2} K_{N} (t) d t = 0$ .

1.2F1F2F4

By [F1], the conjugate partial sum is the trigonometric polynomial $C S_{N} f = \sum_{0 < ∣ k ∣ \leq N} (- i sgn k) \hat{f} (k) e_{k}$ . Expanding $K_{N}$ from 1.1 and substituting $u = x - t$ in each finite sum as in [F2] and [F4], $\int_{0}^{1} K_{N} (t) f (x - t) d t = \sum_{0 < ∣ k ∣ \leq N} (- i sgn k) \hat{f} (k) e_{k} (x) = C S_{N} f (x)$ for every $x$ . The bounded finite kernel makes the integral exist for each translate of the $L^{1}$ representative, and the finite coefficient calculation is exact; this keeps the finite- $N$ object a polynomial-level convolution and makes no claim on any cotangent kernel.

2.1step 1.1step 1.2F10

Fix a point $x$ at which some representative of $f$ is $C^{1}$ on an open interval containing $x$ , and put $g (t) : = f (x - t) - f (x)$ for $∣ t ∣ < 1 / 2$ , so that $∣ g (t) ∣ \leq C ∣ t ∣$ for a constant $C$ and all small $t$ by [F10]. Step 1.2 gives $C S_{N} f (x) = \int_{0}^{1} K_{N} (t) f (x - t) d t$ ; since $K_{N}$ is one-periodic, odd, and has $\int_{- 1 / 2}^{1 / 2} K_{N} = 0$ by 1.1, that integral equals $\int_{∣ t ∣ < 1 / 2} K_{N} (t) g (t) d t$ . The closed form of 1.1 splits the kernel as $\cot (π t) - \cos ((2 N + 1) π t) / \sin (π t)$ , so $C S_{N} f (x) = A (x) - R_{N} (x)$ with $A (x) : = \int_{∣ t ∣ < 1 / 2} \cot (π t) g (t) d t$ and $R_{N} (x) : = \int_{∣ t ∣ < 1 / 2} \frac{\cos ((2 N + 1) π t)}{\sin (π t)} g (t) d t$ , the integrands being defined and measurable off the null point $t = 0$ .

2.2step 1.2F6F7F9

Put $a_{k} : = - i sgn (k) \hat{f} (k)$ . Since $∣ a_{k} ∣ \leq ∣ \hat{f} (k) ∣$ for every $k$ (with $a_{0} = 0$ ), [F6] gives $\sum_{k} ∣ a_{k} ∣^{2} \leq ∥ f ∥_{2}^{2} < \infty$ . Thus [F7] supplies a unique class $C f \in L^{2} (T; C)$ whose symmetric partial sums are exactly the $C S_{N} f$ of 1.2 and which is their $L^{2}$ limit; by [F9] there is an increasing sequence $N_{j} \to \infty$ with $C S_{N_{j}} f (x) \to C f (x)$ for almost every $x$ .

3.1step 2.1F8F10F12

For the point $x$ of 2.1, [F12] writes $\cos ((2 N + 1) π t) = \cos (2 π N t) \cos (π t) - \sin (2 π N t) \sin (π t)$ , so $R_{N} (x) = \int_{∣ t ∣ < 1 / 2} \cos (2 π N t) \frac{\cos (π t) g (t)}{\sin (π t)} d t - \int_{∣ t ∣ < 1 / 2} \sin (2 π N t) g (t) d t$ . Both $t \mapsto \cos (π t) g (t) / \sin (π t)$ and $t \mapsto g (t)$ are integrable on $(- 1 / 2, 1 / 2)$ : the second because $f$ is $L^{1}$ on the finite torus and the quotient is bounded near $0$ by $C$ ; away from $0$ , $1 / \sin (π t)$ is bounded and $g \in L^{1}$ , so the quotient is integrable there as well. Extending them by zero to one period, [F8] gives $\hat{h} (N) \to 0$ for these integrable functions, hence $R_{N} (x) \to 0$ as $N \to \infty$ ; the convergence is at every such $x$ , and no uniformity in $x$ is claimed.

3.2step 2.1F10F11

Also at the point $x$ of 2.1, for $0 < ε < 1 / 2$ the oddness of $\cot$ gives $\int_{ε < ∣ t ∣ < 1 / 2} \cot (π t) f (x - t) d t = \int_{ε < ∣ t ∣ < 1 / 2} \cot (π t) g (t) d t$ , and by [F10] the function $\cot (π t) g (t)$ is integrable on $(- 1 / 2, 1 / 2)$ ; [F11] therefore gives $\int_{ε < ∣ t ∣ < 1 / 2} \cot (π t) g (t) d t \to A (x)$ as $ε ↓ 0$ . So the symmetric principal value exists at $x$ and equals $A (x)$ .

4.1step 2.2step 3.1step 3.2∎

Combining 3.1 and 3.2, for every $x$ at which $f$ is $C^{1}$ near $x$ the finite convolutions satisfy $C S_{N} f (x) = A (x) - R_{N} (x) \to A (x)$ , so the full sequence $C S_{N} f (x)$ converges to the principal value at every such $x$ ; by 2.2 it also converges to $C f (x)$ along a subsequence for almost every $x$ . Therefore $C f (x) = \lim_{ε ↓ 0} \int_{ε < ∣ t ∣ < 1 / 2} f (x - t) \cot (π t) d t$ for almost every $x$ in the open set where $f$ is $C^{1}$ near $x$ , as asserted.

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Sources

Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)